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\begin{document}

\Large
\noindent \textbf{Problem Set 3} \\
\normalsize
ME 444: Continuum Mechanics \\
D.~H.\ Kelley \\
\numpoints~points \\

\begin{questions}

        \question Consider the velocity field
            \begin{eqnarray*}
                v_1 & = & -\frac{n \sin nt}{2 + \cos nt} \, x_1 \\
                v_2 & = & \frac{n \cos nt}{2 + \sin nt} \, x_2 \\
                v_3 & = & 0, 
            \end{eqnarray*}
            where $x_i$ are the spatial coordinates, $t$ is time, and $n$ is a scalar constant.
            \begin{parts}
                \part[2] Find expressions for the $x_i$ in terms of $t$ and the material coordinates $X_i$. 
                \part[2] Determine the particle paths~--- that is, combine the $x_i$ into expressions depending on the $X_i$ but not on $t$. 
                \part[2] Determine the streamlines (the Eulerian field lines of the velocity), which occur where 
                    \[ 
                    \frac{dx_1}{v_1} = \frac{dx_2}{v_2} = \frac{dx_3}{v_3}.
                    \]
                \part[2] Find the volume $v$ of a material region that has volume $V$ in the reference configuration. 
            \end{parts}

        \question[6] Consider the ellipsoid given by 
            \[
            \left( \frac{x_1}{a_1} \right)^2 \left( \frac{t}{\tau} \right)^{\alpha_1} +\left( \frac{x_2}{a_2} \right)^2 \left( \frac{t}{\tau} \right)^{\alpha_2} +\left( \frac{x_3}{a_3} \right)^2 \left( \frac{t}{\tau} \right)^{\alpha_3} = 1,
            \]
            where $x_i$ are the spatial coordinates, and $a_1$, $a_2$, $a_3$, $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\tau$ are all constants and $t>0$. Find a velocity field for which the ellipsoid is a material surface (that is, its material derivative is zero). What values must $\alpha_1$, $\alpha_2$, and $\alpha_3$ take for volumes to be preserved? 

    \question[2] Given the velocity field % Lai:2009 3.49
        \begin{eqnarray*}
            v_1 &=&  k (x_2 - 2)^2 x_3 \\
            v_2 &=&  -x_1 x_2 \\
            v_3 &=&  k x_1 x_3,
        \end{eqnarray*}
        for an incompressible fluid in the $\left\{ \hat{\bm{e}}_1, \hat{\bm{e}}_2, \hat{\bm{e}}_3 \right\}$ basis, determine the value of the constant $k$ such that the equation of mass conservation for an incompressible material is satisfied. 

    \question Consider a motion that causes the instantaneous deformation, in the current configuration, to be % adapted from Lai:2009 3.63
        \begin{eqnarray*}
            u_1 &=& 3X_3 \\
            u_2 &=& -X_1 \\
            u_3 &=& -2X_2
        \end{eqnarray*}
        in the $\left\{ \hat{\bm{e}}_1, \hat{\bm{e}}_2, \hat{\bm{e}}_3   \right\}$ basis. Note that the deformation is \emph{not} small. 
        \begin{parts}
            \part[2] Find the ratio of deformed volume to initial volume using the deformation gradient $\bm{F}$.  
            \part[3] Find the ratio of deformed volume to initial volume using the infinitesimal strain $\bm{E}$. Is this value or the one calculated from $\bm{F}$, above, more accurate? 
%             \part[2] Find the infinitesimal rotation tensor $\bm{\Omega}$.  
%             \part[2] Find the left Cauchy-Green strain tensor $\bm{B}$. 
%             \part[2] Find the right Cauchy-Green strain tensor $\bm{C}$. 
            \part[2] Find the elongation (that is, the ratio of its length in the current configuration to its length in the reference configuration) of a material element along the $\hat{\bm{e}}_3 + \hat{\bm{e}}_1$ direction in the reference configuration, using the deformation gradient. 
            \part[3] Find the elongation of a material element along the $\hat{\bm{e}}_3 + \hat{\bm{e}}_1$ direction in the reference configuration, using the infinitesimal strain. Is this value or the one calculated from $\bm{F}$, above, more accurate? 
            \part[2] Find the current angle between material elements along the $\hat{\bm{e}}_3 + \hat{\bm{e}}_1$ and $2\hat{\bm{e}}_3 + \hat{\bm{e}}_2$ directions in the reference configuration, using the deformation gradient.  
            \part[3] Find the current angle between material elements along the $\hat{\bm{e}}_3 + \hat{\bm{e}}_1$ and $2\hat{\bm{e}}_3 + \hat{\bm{e}}_2$ directions in the reference configuration, using the infinitesimal strain. Is this value or the one calculated from $\bm{F}$, above, more accurate? 
        \end{parts}

    \question[2] The stress distribution in a certain body is given by % Lai:2009 4.4
        \[ \bm{T} = 
        \left[
        \begin{array}{ccc}
            0 & 100x_1 & -100x_2 \\
            100x_1 & 0 & 0 \\
            -100x_2 & 0 & 0
        \end{array}
        \right] \mathrm{MPa} \] 
        in Cartesian coordinates $(x_1,x_2,x_3)$. Find the stress vector acting on a plane that passes through the point $\bm{x}=[1/2, \sqrt{3}/2, 3]$ and is tangent to the circular cylindrical surface $x_1^2 + x_2^2 = 1$. 

    \question Consider the following stress distribution: % Lai:2009 4.7
        \[ \bm{T} = 
        \left[
        \begin{array}{ccc}
            \alpha x_2 & \beta & 0 \\
            \beta & 0 & 0 \\
            0 & 0 & 0
        \end{array}
        \right],
        \]
        where $\alpha$ and $\beta$ are constants. 
        \begin{parts}
            \part[2] Determine the stress vector acting on the square in the $x_1=0$ plane with vertices located at $(0,1,1)$, $(0,-1,1)$, $(0,1,-1)$, and $(0,-1,-1)$.  Sketch each of its components. 
            \part[2] Find the total resultant force of the stress vectors acting on the square. 
            \part[2] Find the total resultant moment about the origin of the stress vectors acting on the square. 
        \end{parts}

    \question For any stress state $\bm{T}$ we define the deviatoric stress $\bm{S}$ to be $\bm{S} = \bm{T} - \frac{1}{3} \left( \tr \bm{T} \right) \bm{1}$, where $\tr \bm{T}$ is the first invariant of the stress tensor $\bm{T}$. % Lai:2009 4.12
        \begin{parts}
            \part[2] Show that the first invariant of the deviatoric stress vanishes.  
            \part[2] Evaluate $\bm{S}$ for the stress tensor 
                \[ \bm{T} = 
                100 \left[
                \begin{array}{ccc}
                    6 & 5 & -2 \\
                    5 & 3 & 4 \\
                    -2 & 4 & 9
                \end{array}
                \right]~\mathrm{kPa}.
                \]
            \part[2] Show that the principal directions of the stress tensor coincide with those of the deviatoric stress tensor. 
        \end{parts}

    \question[2] Consider the stress tensor % Lai:2009 4.31
        \[ \bm{T} = \alpha
        \left[
        \begin{array}{ccc}
            x_2 & -x_3 & 0 \\
            -x_3 & 0 & -x_2 \\
            0 & -x_2 & T_{33}
        \end{array}
        \right].
        \]
        Find an expression for $T_{33}$ such that the stress tensor satisfies the equations of equilibrium in the presence of the body force $\bm{b} = -g \hat{\bm{e}}_3$, where $g$ is a constant. That is, find the equilibrium stress in the presence of gravity. 

    \question Given % Lai 4.21
        \[ \bm{T} = \left[ \begin{array}{ccc}
            0 & 100 & 0 \\
            100 & 0 & 0 \\
            0 & 0 & 0
        \end{array} \right] \, \mathrm{MPa}, \]
        \begin{parts}
            \part[2] Determine the magnitude of shear stress on the plane whose normal is in the direction of $\bm{e}_1 + \bm{e}_2$. 
            \part[4] Determine the maximum and minimum normal stresses and the planes on which they act. 
            \part[3] Determine the maximum shear stress and the plane on which it acts. 
        \end{parts}

    \question[3] In the absence of body forces, the equilibrium stress distribution for a certain body is % Lai:2009 4.32
        \begin{eqnarray*}
            T_{11} &=& A x_2, \\
            T_{12} = T_{21} &=& x_1, \\
            T_{22} &=& B x_1 + C x_2, \\
            T_{33} &=& \frac{1}{2} \left( T_{11}+T_{22} \right),
        \end{eqnarray*}
        and all other $T_{ij}=0$. Also, the boundary plane $x_1-x_2=0$ for the body is free of stress. Find $A$, $B$, and $C$. 

    \question[2] What topic on this problem set was easy? What topic was hard? What topic was impossible? 

\end{questions}
\end{document}